Unit 3: Absolute Value

Transcription

1 Name: Block: Unit 3: Absolute Value Day 1: Characteristics of Absolute Value Day 2: Transformations of Absolute Value Day 3: Absolute Value Equations Day 4: Absolute Value Inequalities 1

2 DAY1: The Absolute Value Function We will: I will: Let s take a look at y = What happens if we change every negative y-value to a positive value? Does this sound familiar? What operation takes negative values and makes them positive? Introducing.. the Absolute Value Function, 0 =, < 0 TURN&TALK: Why does the absolute value function look like a V instead of a U? We can analyze the parent function for special points and behavior.. y= Domain: Range: Verte: y-intercept: zeros (roots, -intercepts, solutions): Increasing: Decreasing: End Behavior: Slope of right branch: 2

3 1. y= 4 2. y= + 2 Domain: Range: Verte: y-intercept: zeros (roots, -intercepts, solutions): Increasing: Decreasing: End Behavior: Domain: Range: Verte: y-intercept: zeros (roots, -intercepts, solutions): Increasing: Decreasing: End Behavior: Slope of right branch: Slope of right branch: 3. y 3 4. = + y= Domain: Range: Verte: y-intercept: zeros (roots, -intercepts, solutions): Increasing: Decreasing: End Behavior: Slope of right branch: Domain: Range: Verte: y-intercept: zeros (roots, -intercepts, solutions): Increasing: Decreasing: End Behavior: Slope of right branch: 3

4 5. y= y= Domain: Range: Verte: y-intercept: zeros (roots, -intercepts, solutions): Increasing: Decreasing: End Behavior: Domain: Range: Verte: y-intercept: zeros (roots, -intercepts, solutions): Increasing: Decreasing: End Behavior: Slope of right branch: Slope of right branch: TURN&TALK: What patterns do you notice as you analyze this function family? Graph the inverse of the Absolute Value Function (start out with the original fn y= ) Is the inverse a function? 4

5 We will: Alg2 Day2: Graphing Using TRANSFORMATIONS I will: Note graph your original function in colored pencil Eploration of Transformations Vertical Shifts 1. Graph y = on your calculator in Y1. a) Sketch a graph of the function. b) What is the verte of the graph? 2. Graph y = + 2 on your calculator in Y2. b) How does the graph move? (up or down) c) What is the verte of the graph? 3. Graph y = - 5 on your calculator in Y2. b) How does the graph move? c) What is the verte of the graph? 4. Graph y = - 1 on your calculator in Y2. b) How does the graph move? c) What is the verte of the graph? 5. Graph y = + 4 on your calculator in Y2. b) How does the graph move? c) What is the verte of the graph? 6. Given that y = a h + k is the symbolic form of the absolute value function, what does the parameter k control? If k is positive, what direction do we move? If k is negative, what direction do we move? 5

6 Eploration of Transformations Horizontal Shifts 1. Graph y = on your calculator in Y1. a) Sketch a graph of the function. b) What is the verte of the graph? 2. Graph y = - 1 on your calculator in Y2. b) How does the graph move? Left or Right? c) What was the SIGN inside the absolute value? d) What is the verte of the graph? 3. Graph y = - 5 on your calculator in Y2. b) How does the graph move? Left or Right? c) What was the SIGN inside the absolute value? d) What is the verte of the graph? 4. Graph y = + 3 on your calculator in Y2. b) How does the graph move? Left or Right? c) What was the SIGN inside the absolute value? d) What is the verte of the graph? 5. Graph y = + 7 on your calculator in Y2. b) How does the graph move? Left or Right? c) What was the SIGN inside the absolute value? d) What is the verte of the graph? 6. Given that y = a h + k is the symbolic from of the absolute value function, what does the parameter h control? When we have h, what direction does the graph move? When we have + h, what direction does the graph move? How is the motion related to the sign of h? 6

7 Eploration of Transformations Vertical Stretch or Shrink 1. Graph y = on your calculator in Y1. a) What direction does the graph open? b) What is the verte of the graph? c) Fill in the table to the right. These coordinates are the basic ordered pairs of the absolute value function. 2. Graph y = 2 on your calculator in Y2. a) What direction does the graph open? b) What is the verte of the graph? c) Fill in the table. How do these y-coordinates compare with the y- coordinates in question 1? Is the graph fatter or skinnier? 3. Graph y = ½ on your calculator in Y2. a) What direction does the graph open? b) What is the verte of the graph? c) Fill in the table. How do these y-coordinates compare with the y- coordinates in question 1? Is the graph fatter or skinnier? y y y 4. Graph y = - on your calculator in Y2. b) How did the graph change? 5. Graph y = -2 on your calculator in Y2. b) How did the graph change? 6. Given that y = a h + k is the symbolic from of the absolute value function, what does the parameter a control? 7

8 Eploration of ALL Transformations 1. Graph y = on your calculator in Y1. 2. Graph y = on you calculator in Y2. a) What direction does the graph open? b) How does the graph move? (left/right, up/down) c) What is the verte of the graph? 3. Graph y = -½ on you calculator in Y2. a) What direction does the graph open? b) How does the graph move? (left/right, up/down) c) What is the verte of the graph? 4. Graph y = on you calculator in Y2. a) What direction does the graph open? b) How does the graph move? (left/right, up/down) c) What is the verte of the graph? 5. Graph y = on you calculator in Y2. a) What direction does the graph open? b) How does the graph move? (left/right, up/down) c) What is the verte of the graph? Given the absolute value function y = a h + k If a > 0, does the graph open up or down? If a < 0, does the graph open up or down? If a > 1, does the graph have a vertical stretch or vertical shrink? If 0 < a < 1, does the graph have a vertical stretch or vertical shrink? What does the parameter k control? What does the parameter h control? What is the verte? Now generalize Fill in the table using your knowledge of transformations. Function 1. y = ¼ y = y = y = - ½ y = y = Direction/Opening (up or down) Verte Vertical Stretch or shrink

9 Graph the functions #1-3 above using your knowledge of transformations y 10 y 10 y Domain: Domain: Domain: Range: Range: Range: Given the absolute value equation graph, write the absolute value equation: 1. y 2. y y y 4. 9

10 Day3: Absolute Value Equations We will: I will: Absolute Value means Absolute Value Equations What it MEANS: Graph an Absolute Value Equation on a Number Line 1. = 4 As distance: - 0 = 4 the set of points whose distance from 0 is equal to 4 Another way these are two FUNCTIONS. Where are they EQUAL? graph y= and y=4 2. = 2 As distance: - 0 = 2 the set of points whose distance from is equal to As functions - What two functions are we looking at here? Where are they EQUAL? graph y= and y= = 3 As distance: the set of points whose distance from is equal to As functions - What two functions are we looking at here? Where are they EQUAL? graph y= -4 and y= = 3 the set of points whose distance from is equal to 10

11 How we DO it ALGEBRAICALLY: Solve an Absolute Value Equation 1. Isolate the absolute value symbol on one side of the equal sign 2. Break the equation into 2 derived equations the positive case and the negative case 3. Solve both equations 4. Check your solutions (WARNING: There may be etraneous solutions!) = = 7 TURN&TALK: Compare/contrast solving absolute value equations with solving linear equations = = = = =

12 TURN&TALK: Given the equation a + b = c, describe the values of c that would yield two solutions, one solution, and no solutions. Discuss this with your partner and write your answer. Review - Solving Linear Inequalities (This is a lead-in to Absolute Value inequalities.) A) Inequality Symbols: <, Inequality Symbols: <, >,,or >, Don t forget switch the sign of the inequality when multiplying or dividing by a negative # Switch Don t switch -3 < 9 3 < -12 Original Problem(s) > -3 < -4 Solution B) Graphing Linear Inequalities: Closed circle Open Circle, <, > C) Solve the following linear inequalities, then graph each solution: EX 1] < 9 EX 2] EX 3] > 4 EX 4]

13 D) Graph Compound Inequalities What is different now? EX] -1 < < 2 EX] -2 or > 1 HOW DO I REMEMBER THESE??? Graph the following inequalities. 1. 3< or > 7 Solve the compound inequality, and then graph your solution < < 10 or 2 4> 4 Important things to remember about solving and graphing inequalities: ***When to use open circle vs. closed circle ***When to switch signs when solving *** hand shake *** *** boat oars *** 13

14 Solve the following compound inequalities, then graph each solution: EX 1] < 14 EX 2] < -13 OR 5 5 > -5 14

15 Day4: Absolute Value Inequalities We will: I will: KEY: Absolute value turns simple inequalities into compound inequalities because we have to consider the negative case. Less Than: Greater Than: < 3 means > 3 means set of pts whose distance from is to set of pts whose distance from is to -3 < < 3 < 3 or > 3 Write the absolute value inequalities that would correspond with these graphs: How we DO it: Solve an Absolute Value Inequalities 6. Isolate the absolute value symbol on one side of the equal sign 7. Break the equation into derived equations the positive case and the negative case (for the negative case KEEP, CHANGE, CHANGE) 8. Solve both equations 9. Check your solutions (WARNING: There may be etraneous solutions!) Solve and Graph the Absolute Value Inequalities Think: 3-2 > 18 Think: 3 2 > 18 OR 3 2 <

16 2 > 4 Think: Think: How about LESS THAN? Think: Think: 16

17 1 < 4 Think: Think: Try these: < < -8 THINK about this one! TURN&TALK: Change something about #3 so that.. 17